C₂₉×#2078 #16815792

order	3712
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₉×#2078

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
prime radical	C₂×#41×C₂₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ G

198 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29)(30,31)(32,35)(33,34)(36,37)(38,39,41,46,43,47,42,45)(40,44,49,52,50,51,48,53);(31,34)(35,37)(38,40)(39,44)(41,49)(42,48)(43,50)(45,53)(46,52)(47,51);(30,32,33,36)(31,35,34,37)(39,45)(40,48)(41,42)(44,51)(46,47)(49,50)

Conjugacy classes & character table

754 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12284, C₂₉×#12285, C₂₉×#12286, …
maximal normal subgroup of	C₂₉×#12284, C₂₉×#12285, C₂₉×#12286, …
maximal quotient of	C₂₉×#10615, C₂₉×#10616, C₂₉×#10617, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 810);